Recall that there is a birational map from \(\Sigma _r\)  to \(\Sigma _{r+1}\) , called an ”elementary transform" (see [1]}), constructed as follows. We choose some fiber \(F\) ,
and consider the surface \(\Sigma _r’\) , the blow-up of \(\Sigma _r\)  at \(p := F \cap E_r\) . Let \(F’, E_r’\)  the
strict transforms of \(F, E_r\)  and \(\tilde{E}\)  be the exceptional divisor of this blow-up. Then we have
\(0 = F^2 = (F’+ \tilde{E})^2 = (F’)^2 + 2 -1\) 
